%0 Journal Article %T A new kind of repairable system with repairman vacations %A Zhang, Xiao %A Guo, Lina %J Journal of Nonlinear Sciences and Applications %D 2015 %V 8 %N 4 %@ ISSN 2008-1901 %F Zhang2015 %X In this paper, a new kind of series repairable system with repairman vacation is discussed, in which the failure rate functions of all the units and the delayed vacation rate function of the repairman are related to the working time of the system. The system model of a group of integro-differential equations is established by using probability analysis method, which then is translated into an initial value problem of a class of abstract semi-linear evolution equation in a suitable Banach space for further study. Then the conditions of the existence and uniqueness of the system solution is analyzed by using \(C_0\)-semigroup theory. And by using Laplace transform method, some steady-state reliability indexes, such as system availability, failure frequency, and the probability that the repairman is on vacation, are discussed. Numerical examples are also presented at the end of the paper. %9 journal article %R 10.22436/jnsa.008.04.05 %U http://dx.doi.org/10.22436/jnsa.008.04.05 %P 324--333 %0 Book %T Introduction to reliability mathematics %A J. H. Cao %A K. Cheng %D 1986 %I Higher education press %C Beijing %F Cao1986 %0 Journal Article %T Stability analysis of a new kind n-unit series repairable system %A L. N. Guo %A H. B. Xu %A C. Gao %A G. T. Zhu %J Appl. Math. Model. %D 2011 %V 35 %F Guo2011 %0 Journal Article %T Bilevel control of degraded machining system with warm standbys, setup and vacation %A M. Jain %A Rakhee %A M. Singh %J Appl. Math. Model. %D 2004 %V 28 %F Jain2004 %0 Journal Article %T A replacement policy for a repairable system with its repairman having multiple vacations %A J. S. Jia %A S. M. Wu %J Comput. Ind. Eng. %D 2009 %V 57 %F Jia2009 %0 Journal Article %T Vacation policies for machine repair problem with two type spares %A J. C. Ke %A K. H. Wang %J Appl. Math. Model. %D 2007 %V 31 %F Ke2007 %0 Journal Article %T The positive solutions of abstract semilinear evolution equations and their applications %A Y. X. Li %J Acta Math. Sinica %D 1996 %V 39 %F Li 1996 %0 Journal Article %T Existence and uniqueness of positive periodic solutions for abstract semilinear evolution equations %A Y. X. Li %J J. Syst. Sci. Math. Sci. %D 2005 %V 28 %F Li 2005 %0 Journal Article %T Existence of Solutions of Initial Value Problems for Abstract Semilinear Evolution Equations %A Y. X. Li %J Acta Math. Sinica %D 2005 %V 48 %F Li 2005 %0 Journal Article %T One-unit repairable system with multiple delay vacations %A R. B. Liu %A Y. H. Tang %J Chinese J. Eng. Math. %D 2006 %V 23 %F Liu2006 %0 Journal Article %T Reliability analysis of a two-dissimilar-unit cold standby system with separated repair rule %A R. B. Liu %A Y. H. Tang %J J. Syst. Eng. %D 2006 %V 21 %F Liu2006 %0 Journal Article %T A new kind of N-unit series repairable system and its reliability analysis %A R. B. Liu %A Y. H. Tang %A C. Y. Luo %J Math. Appl. %D 2007 %V 20 %F Liu2007 %0 Book %T Semigroups of linear operators and applicaton to partial differential equations %A A. Pazy %D 1983 %I Springer-Verlag %C New York %F Pazy1983 %0 Book %T Studies of some aspects on abstract semilinear evolution solutions and functional differential equations %A L. L. Wang %D 2001 %I College of Mathematics and Econometrics %C Hunan University %F Wang2001 %0 Journal Article %T Reliability analysis of two-unit cold standby repairable systems under Poisson shocks %A Q. T. Wu %A S. M. Wu %J Appl. Math. Comput. %D 2011 %V 218 %F Wu2011 %0 Journal Article %T A deteriorating system with its repairman having multiple vacations %A L. Yuan %A J. Xu %J Appl. Math. Comput. %D 2011 %V 217 %F Yuan2011